Improved distance scaling for sparsified FDPC ensembles

Establish an improved asymptotic minimum-distance scaling for the ensemble of L-sparsified order-s FDPC codes by exploiting the additional parity constraints introduced through sparsification, beyond the minimum-distance guarantee inherited from the parent FDPC code.

Background

The qFDPC construction obtains its distance guarantee from the parent order-s FDPC ensemble. Although L-sparsification adds parity constraints and can suppress low-weight codewords, the paper's theorem uses only the fact that sparsification produces a subcode of the parent code, so it does not improve the parent-code distance bound.

The paper's weight-distribution analysis suggests that sparsification may substantially reduce the multiplicity of low-weight codewords. A sharper asymptotic analysis of the sparsified ensemble could therefore yield a stronger distance exponent or otherwise improve the currently established scaling.

References

The distance bound in Theorem~\ref{thm:qfdpc-scaling} is conservative in the sense that it only uses the minimum-distance guarantee of the parent FDPC code. In particular, the additional parity constraints introduced by $L$-sparsification are used to control the rate and stabilizer weight, but their effect on the minimum distance is not exploited. The weight-distribution analysis of Section~\ref{sec:fdpc-extension} suggests that sparsification can substantially suppress low-weight codewords. Therefore, sparsification may yield stronger distance scaling than the parent-code guarantee alone. Establishing an improved asymptotic distance scaling for the sparsified ensemble is therefore left for future work.

— Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes  (2609.01181 - Mahdavifar, 1 Sep 2026) in Remark 1, Section 4.2 (Quantum FDPC Codes)

Several directions remain open. Deterministic choices of the sparsification and underlying permutations in the FDPC construction, together with a sharper characterization of the good-distance subensemble, could further improve the finite-length constructions and their ML estimates.

— Quantum LDPC and High-Rate CSS Codes from Fair-Density Parity-Check Codes  (2609.01181 - Mahdavifar, 1 Sep 2026) in Section 7 (Conclusion)

Can the sparse protected construction reach linear distance, or select a useful fixed logical sector while retaining sparse checks?

— Constant Rate Codes with Fully Addressable Transversal T: Good Codes, Sparse Checks  (2609.38996 - Meiburg, 30 Sep 2026) in Section 6, Future Work